Problem
Objects versus gaps when spacing along a line
Along one side of a straight road 15 m long, trees are planted from the very start to the very end, spaced 3 m apart. I need to find how many trees there are in all.
Operations
Your answer
trees
Watch out It's tempting to stop at the number of gaps (5) — but since both ends have a tree, the number of trees is one more than the number of gaps.
How to solve
Strategy Draw a Diagram — Spacing-along-a-line problems are clearest with a diagram: drawing the road and marking trees shows that the number of gaps and the number of trees are not the same. First find how many 3 m gaps fit in 15 m (a division subproblem), then add 1 because a line with trees at both ends has one more tree than it has gaps.
1STEP 1
Count the equal gaps along the road
Each gap between neighboring trees is 3 m. Divide the road length by the gap length to find how many gaps there are: 5 gaps.
15 ÷ 3 = 5
Fitting equal lengths into a total length is a division: how many 3 m pieces make 15 m.
3.OA.A.3Identify Subproblems2STEP 2
Turn gaps into trees using the diagram
On a straight road with a tree at both the start and the end, there is always one more tree than the number of gaps: 5 + 1 = 6.
5 + 1 = 6
Picture fence posts: 5 spans between posts means 6 posts, because the two endpoints each carry a post.
3.OA.D.8Draw A DiagramOn a straight road with a tree at both the start and the end, the number of trees is one more than the number of gaps.
Why?
The trees split into two groups that never overlap — the tree that closes each gap, and the single starting tree that closes no gap — so adding the two counts gives the total number of trees.
Why?
Every gap is closed by exactly one tree at its far end, and different gaps end at different trees, so these end-trees match the gaps one for one and are equal in number to the gaps.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Sorting the trees into the gap-closing trees and the lone starting tree leaves none out and counts none twice, so adding the size of each group accounts for every tree.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Answer
6 trees
5 + 1 = 6
The figure shows 6 trees and 5 gaps, matching the count. Checking the length: 5 gaps of 3 m each is 5 times 3 equals 15 m, exactly the road length, so the trees reach from start to end with none left over.
Takeaway
Count the gaps, then add one for the extra end tree: Grade 3 division plus a fence-post idea!
- Count the equal gaps along the road
- Turn gaps into trees using the diagram
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested